These equations have 2nd derivatives because acceleration is in Newton's Law F = ma. The key model equation is (second derivative) y'' = minus y or y'' = minus a^2 y.
There are two solutions since the equation is second order. They are sine and cosine.
y = sin (at) and y = cos (at). Two derivatives bring back sine and cosine with minus a^2. 
The next step allows damping (first derivative) as in my'' + dy' + ky = 0. How to solve? Just try y = e^at. You find that ma^2 + da + k = 0. Two a's give two solutions. 

1. Faculty Introduction
2. Big Picture of Calculus
3. Big Picture: Derivatives
4. Max and Min and Second Derivative
5. The Exponential Function
6. Big Picture: Integrals
7. Derivative of sin x and cos x
8. Product Rule and Quotient Rule
9. Chains f(g(x)) and the Chain Rule
10. Limits and Continuous Functions
11. Inverse Functions f ^-1 (y) and the Logarithm x = ln y
12. Derivatives of ln y and sin ^-1 (y)
13. Growth Rate and Log Graphs
14. Linear Approximation/Newton's Method
15. Power Series/Euler's Great Formula
16. Differential Equations of Motion
17. Differential Equations of Growth
18. Six Functions, Six Rules, and Six Theorems

Highlights of Calculus is a series of short videos that introduces the basics of calculus—how it works and why it is important. The intended audience is high school students, college students, or anyone who might need help understanding the subject.

The series is divided into three sections:
Introduction
- Why Professor Strang created these videos
- How to use the materials

Highlights of Calculus
- Five videos reviewing the key topics and ideas of calculus
- Applications to real-life situations and problems
- Additional summary slides and practice problems

Derivatives
- Twelve videos focused on differential calculus
- More applications to real-life situations and problems
- Additional summary slides and practice problems

Acknowledgements
Special thanks to Professor J.C. Nave for his help and advice on the development and recording of this program.The video editing was funded by the Lord Foundation of Massachusetts.